I suspect it goes along the lines of: show that there is proper class of ordinals, and show that for each ordinal one can construct inequivalent surreal numbers, or something.
Since there are 'more' ordinals than cardinals, and there is a proper class of cardinals (if there is a set of cardinals, C, what is the cardinality of the power set of C?), this becomes easy - if I've understood the one thing I read about surreal numbers and some link to ordinals, and that;s a big if.
I don't understand why the surreal numbers don't form a complete ordered field, in which case the real numbers wouldn't be the only complete ordered field. I guess technically if the surreals form a proper class then it wouldn't count, but that's... wrong, somehow.
I don't know what you've read so I can't really comment, but completeness usually requires some notion of a metric - i.e. a mapping into R, the real numbers. So what is the norm of any non-real surreal?
(Rehash of my earlier post... only this time, correct)
Ignoring that the surreals form a proper class rather than a set, you'd need to prove that you can't define a metric on them that turns them into a complete ordered field.
One way to see that the surreal numbers are not a complete ordered field is to use the axioms of a complete ordered field to show that they satisfy the archimedean property, which says that there are no infinitesimals other than 0 and no infinities. Since the surreal numbers have non-zero infinitesimals and infinities, they cannot be a complete ordered field.